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## Probability, Uncertainty and Quantitative Risk

January 2017 , Volume 2

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2017, 2: 14
doi: 10.1186/s41546-017-0026-3

*+*[Abstract](546)*+*[HTML](251)*+*[PDF](670.26KB)**Abstract:**

We study the concept of financial bubbles in a market model endowed with a set $\mathcal{P}$ of probability measures, typically mutually singular to each other. In this setting, we investigate a dynamic version of robust superreplication, which we use to introduce the notions of bubble and robust fundamental value in a way consistent with the existing literature in the classical case $\mathcal{P}$={$\mathbb{P}$}. Finally, we provide concrete examples illustrating our results.

2017, 2: 13
doi: 10.1186/s41546-017-0024-5

*+*[Abstract](549)*+*[HTML](234)*+*[PDF](892.85KB)**Abstract:**

We study robust notions of good-deal hedging and valuation under combined uncertainty about the drifts and volatilities of asset prices. Good-deal bounds are determined by a subset of risk-neutral pricing measures such that not only opportunities for arbitrage are excluded but also deals that are too good, by restricting instantaneous Sharpe ratios. A non-dominated multiple priors approach to model uncertainty (ambiguity) leads to worst-case good-deal bounds. Corresponding hedging strategies arise as minimizers of a suitable coherent risk measure. Good-deal bounds and hedges for measurable claims are characterized by solutions to secondorder backward stochastic differential equations whose generators are non-convex in the volatility. These hedging strategies are robust with respect to uncertainty in the sense that their tracking errors satisfy a supermartingale property under all a-priori valuation measures, uniformly over all priors.

2017, 2: 12
doi: 10.1186/s41546-017-0023-6

*+*[Abstract](514)*+*[HTML](232)*+*[PDF](651.78KB)**Abstract:**

We develop a dynamic optimization framework to assess the impact of funding costs on credit swap investments. A defaultable investor can purchase CDS upfronts, borrow at a rate depending on her credit quality, and invest in the money market account. By viewing the concave drift of the wealth process as a continuous function of admissible strategies, we characterize the optimal strategy in terms of a relation between a critical borrowing threshold and two solutions of a suitably chosen system of first order conditions. Contagion effects between risky investor and reference entity make the optimal strategy coupled with the value function of the control problem. Using the dynamic programming principle, we show that the latter can be recovered as the solution of a nonlinear HJB equation whose coeffcients admit singular growth. By means of a truncation technique relying on the locally Lipschitzcontinuity of the optimal strategy, we establish existence and uniqueness of a global solution to the HJB equation.

2017, 2: 11
doi: 10.1186/s41546-017-0022-7

*+*[Abstract](719)*+*[HTML](397)*+*[PDF](566.92KB)**Abstract:**

One of the fundamental issues in Control Theory is to design feedback controls. It is well-known that, the purpose of introducing Riccati equations in the study of deterministic linear quadratic control problems is exactly to construct the desired feedbacks. To date, the same problem in the stochastic setting is only partially well-understood. In this paper, we establish the equivalence between the existence of optimal feedback controls for the stochastic linear quadratic control problems with random coefficients and the solvability of the corresponding backward stochastic Riccati equations in a suitable sense. We also give a counterexample showing the nonexistence of feedback controls to a solvable stochastic linear quadratic control problem. This is a new phenomenon in the stochastic setting, significantly different from its deterministic counterpart.

2017, 2: 10
doi: 10.1186/s41546-017-0017-4

*+*[Abstract](537)*+*[HTML](248)*+*[PDF](548.71KB)**Abstract:**

This paper provides sufficient conditions for the time of bankruptcy (of a company or a state) for being a totally inaccessible stopping time and provides the explicit computation of its compensator in a framework where the flow of market information on the default is modelled explicitly with a Brownian bridge between 0 and 0 on a random time interval.

2017, 2: 9
doi: 10.1186/s41546-017-0020-9

*+*[Abstract](743)*+*[HTML](422)*+*[PDF](1891.72KB)**Abstract:**

The paper presents a comprehensive model of a banking system that integrates network effects, bankruptcy costs, fire sales, and cross-holdings. For the integrated financial market we prove the existence of a price-payment equilibrium and design an algorithm for the computation of the greatest and the least equilibrium. The number of defaults corresponding to the greatest price-payment equilibrium is analyzed in several comparative case studies. These illustrate the individual and joint impact of interbank liabilities, bankruptcy costs, fire sales and cross-holdings on systemic risk. We study policy implications and regulatory instruments, including central bank guarantees and quantitative easing, the significance of last wills of financial institutions, and capital requirements.

2017, 2: 8
doi: 10.1186/s41546-017-0021-8

*+*[Abstract](451)*+*[HTML](239)*+*[PDF](1444.31KB)**Abstract:**

Risks embedded in asset price dynamics are taken to be accumulations of surprise jumps. A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level. Estimation is done by matrix exponentiation of the transition rate matrix for a continuous time finite state Markov chain approximation. The motion is decomposed into a space dependent drift and a space dependent martingale component. Though there is some local mean reversion in the drift, space dependence of the martingale component renders the dynamics to be of the momentum type. Local risk is measured using market calibrated measure distortions that introduce risk charges into the lower and upper prices of two price economies. These risks are compensated by the exponential variation of space dependent arrival rates. Estimations are conducted for the S&P 500 index (

*SPX*), the exchange traded fund for the financial sector (

*XLF*), J. P. Morgan stock prices (

*JPM*), the ratio of

*JPM*to

*XLF*, and the ratio of

*XLF*to

*SPX*.

2017, 2: 7
doi: 10.1186/s41546-017-0019-2

*+*[Abstract](568)*+*[HTML](232)*+*[PDF](939.57KB)**Abstract:**

We apply to the concrete setup of a bank engaged into bilateral trade portfolios the XVA theoretical framework of Albanese and Crépey (2017), whereby so-called contra-liabilities and cost of capital are charged by the bank to its clients, on top of the fair valuation of counterparty risk, in order to account for the incompleteness of this risk. The transfer of the residual reserve credit capital from shareholders to creditors at bank default results in a unilateral CVA, consistent with the regulatory requirement that capital should not diminish as an effect of the sole deterioration of the bank credit spread. Our funding cost for variation margin (FVA) is defined asymmetrically since there is no benefit in holding excess capital in the future. Capital is fungible as a source of funding for variation margin, causing a material FVA reduction. We introduce a specialist initial margin lending scheme that drastically reduces the funding cost for initial margin (MVA). Our capital valuation adjustment (KVA) is defined as a risk premium, i.e. the cost of remunerating shareholder capital at risk at some hurdle rate.

2017, 2: 6
doi: 10.1186/s41546-017-0018-3

*+*[Abstract](671)*+*[HTML](319)*+*[PDF](599.16KB)**Abstract:**

In this introductory paper to the issue, I will travel through the history of how quantitative finance has developed and reached its current status, what problems it is called to address, and how they differ from those of the pre-crisis world.

2017, 2: 5
doi: 10.1186/s41546-017-0016-5

*+*[Abstract](577)*+*[HTML](233)*+*[PDF](470.3KB)**Abstract:**

We show the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite

*q*-variation with

*q*∈[1, 2). In contrast to previous work, we apply a direct fixpoint argument and do not rely on any type of flow decomposition. The resulting object is an effective tool to study semilinear rough partial differential equations via a Feynman-Kac type representation.

2017, 2: 4
doi: 10.1186/s41546-017-0013-8

*+*[Abstract](579)*+*[HTML](244)*+*[PDF](621.94KB)**Abstract:**

G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation. To prove the self-normalized central limit theorem, we also establish a new Donsker's invariance principle with the limit process being a generalized G-Brownian motion.

2017, 2: 3
doi: 10.1186/s41546-017-0012-9

*+*[Abstract](680)*+*[HTML](405)*+*[PDF](1368.54KB)**Abstract:**

In this work we give a comprehensive overview of the time consistency property of dynamic risk and performance measures, focusing on a the discrete time setup. The two key operational concepts used throughout are the notion of the LMmeasure and the notion of the update rule that, we believe, are the key tools for studying time consistency in a unified framework.

2017, 2: 2
doi: 10.1186/s41546-017-0015-6

*+*[Abstract](479)*+*[HTML](238)*+*[PDF](590.46KB)**Abstract:**

Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate. By their definition, they imply a unique probability density function. The applications of default probability distributions are varied, including the risk premium model used to price default bonds, reliability measurement models, insurance, etc. Fractional probability density functions (FPD), however, are not in general conventional probability density functions (Tapiero and Vallois, Physica A,. Stat. Mech. Appl. 462:1161-1177, 2016). As a result, a fractional FPD does not define a fractional hazard rate. However, a fractional hazard rate implies a unique and conventional FPD. For example, an exponential distribution fractional hazard rate implies a Weibull probability density function while, a fractional exponential probability distribution is not a conventional distribution and therefore does not define a fractional hazard rate. The purpose of this paper consists of defining fractional hazard rates implied fractional distributions and to highlight their usefulness to granular default risk distributions. Applications of such an approach are varied. For example, pricing default bonds, pricing complex insurance contracts, as well as complex network risks of various granularity, that have well defined and quantitative definitions of their hazard rates.

2017, 2: 1
doi: 10.1186/s41546-017-0014-7

*+*[Abstract](781)*+*[HTML](468)*+*[PDF](594.73KB)**Abstract:**

In this paper, we study the recursive stochastic optimal control problems. The control domain does not need to be convex, and the generator of the backward stochastic differential equation can contain

*z*. We obtain the variational equations for backward stochastic differential equations, and then obtain the maximum principle which solves completely Peng's open problem.

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